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On the spectrum of the Cesaro operator. (English) Zbl 0548.47017
The Cesàro operator \(Cx=y\) where \(y_ n=\frac{x_ 1+x_ 2+...+x_ n}{n}\) is shown to have spectrum \(|\lambda -{1\over2}|\leq {1\over2}\) when acting on the space \(c_ 0\) of sequences convergent to zero. C is shown to have no eigenvalues, whilst its adjoint \(C^*\) has eigenvalues \(|\lambda -{1\over2}| <{1\over2}\) all simple. The methods are similar to those of Halmos et. al. dealing with C on \(\ell^ 2\) (reference given) except that a direct proof of the invertibility of C-\(\lambda\) I for \(|\lambda -{1\over2}| >{1\over2}\) is needed.

MSC:
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
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